The base of a prism increases by 20%, by what percent must the height be decreased so as to maintain the same volume

Questions : The base of a prism increases by 20%, by what percent must the height be decreased so as to maintain the same volume ?

Answer:  

 Here's a shortcut formula for percentage change problems like this:

$$\text{Required Percentage Decrease} = \frac{\text{Increase Percentage}}{100 + \text{Increase Percentage}} \times 100$$

Substituting the given increase of 20%:

$$= \frac{20}{100 + 20} \times 100$$ $$= \frac{20}{120} \times 100$$ $$= \frac{1}{6} \times 100$$ $$= 16.67\%$$

So, the height must be decreased by 16.67% to keep the volume constant.

Additional Method 

Yes! Here’s an alternative approach using ratios.

Step 1: Define Variables

Let the original base area be $A$ and the original height be $h$.
The original volume of the prism is:

$$V = A \times h$$

When the base area increases by 20%, the new base area becomes:

$$A' = 1.2A$$

Let the new height be $h'$, and the new volume must remain the same:

$$V = A' \times h'$$

Step 2: Set Up the Equation

Since the volume remains unchanged:

$$A \times h = 1.2A \times h'$$

Dividing both sides by $A$:

$$h = 1.2h'$$

Step 3: Express Height Change as a Percentage

Solving for $h'$:

$$h' = \frac{h}{1.2}$$

The percentage decrease in height is:

$$\frac{h - h'}{h} \times 100$$ $$= \frac{h - \frac{h}{1.2}}{h} \times 100$$

Factor out $h$:

$$= \left( 1 - \frac{1}{1.2} \right) \times 100$$

Approximating:

$$= \left( 1 - 0.8333 \right) \times 100$$ $$= 0.1667 \times 100$$ $$= 16.67\%$$

Conclusion

The height must be decreased by 16.67% to maintain the same volume.

Short Cut 

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